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Show that the series (p^iff + <pA2# + ---- 1- <p*n9 -h • • • converges almost everywhere.
(Observe that

*f- ± U*


almost everywhere; integrate both sides, and use (13.8.5).)

(b)   Let B be the measurable set of points x e X such that (Un - g)(x) > 0 for at

least one integer n ^ 0. Show that in B the sequence of functions

converges to 0 almost everywhere. (Deduce from (a) above that the intersection of
Q An with the set (x e X : g(x) > 0} is negligible. Now replace g and /successively by

Um -g and Um •/, where m ^ 1.)

(c)   Let B0 be the measurable set of points x e X such that g(x) > 0. For each x e B0 „


+ (U < gm + • - • + (t/-1 • g)(x)
and let R*(/, g) = sup | Rn(/, g) \ . Show that R*(/, g)(x) < + oo for almost all x e Ba .


(For each t > 0, let At be the set of points x e B0 such that R*(/, g)(x) > t. Show that
*f   9 dp ^ Ni(/) by applying Problem 16(d) to |/| - t • g.)

(d)   Let $ e «S?i be such that ®(x) > 0 for all x e X (cf. (1 3.1 5.7)). Show that the set
Xoo of points x e X such that

®(x) + (£/• <£)(*) + ••• + (£/"• <&)(*) + ••• = + oo

does not depend on the choice of <1>, but only on U.
For each x e X, let

Gn(x) = $(*) + ( U • <S>)(x) +•
and  let G(x) = lim Gn(x) (so that GW = -f oo for x £ Xoo). Show that for each


integrable set A c: Xoo and each f > 0 we have t - (U • <pA) ^ G, and hence, that the
function U - <pA is zero almost everywhere in X0 = CX°Q . (Observe that, if An is the
set of x e A such that Gn(;c) > t, then *9?An <; Gn .)


If/is a mapping of X into R, the maximum in measure (resp. minimum in
measure) of/on X (with respect to the measure p) is defined to be the greatest
lower bound (resp. least upper bound) of the real numbers a such that
f(x) ^ a (resp. f(x) ^ d) almost everywhere with respect to \JL, and is denoted
by M*>(/), or ess sup/, or ess sup/(x) (resp. mJJ), or ess inf/, or ess inf/(;c)).

Clearly we have m00(/) = -MJ-/).h satisfy the